This retraction corresponds to a second order approximation of the geodesics of \(\mathcal {S}^{++}_p\)  [1]}, which generalize the concept of straight lines for a manifold.
We now have all the necessary elements to perform the optimization of \(f\)  on \(\mathcal {S}^{++}_p\) .
For instance, the sequence of iterates \(\lbrace {\Sigma }_t\rbrace \)  and descent directions \(\lbrace {\xi }_t\rbrace \)  generated by a Riemannian conjugate gradient algorithm is
\(\begin{array}{l}{\Sigma }_{t+1} = R^{\mathcal {S}^{++}_p}_{{\Sigma }_t}({\xi }_t)\\[3pt]{\xi }_t = \alpha _t(-\nabla ^{\mathcal {S}^{++}_p}f({\Sigma }_t) + \beta _t\mathcal {T}^{\mathcal {S}^{++}_p}_{{\Sigma }_{t-1}\rightarrow {\Sigma }_t}({\xi }_{t-1})),\end{array}\) 
